Billeder på siden
PDF
ePub

lid, having the common base, ABC; then, if the folid EABC is not equal to the folid FABC, let it be equal to fome folid as GA BC, either greater or lefs than EABC, which cannot be; for the one would contain the other; and if the folid angle is con

[blocks in formation]

tained by more than three plane angles, equal and fimilar to one another, then it can be divided into angles which are contained by three equal and fimilar plane angles, by Prop. 20. Book VI. and parts have the fame proportion C their like multiples, by Prop 15. Book V.. wherefore univerfally, figures bounded by an equal number of

equal and fimilar planes are equal and fimilar.

as

N. B. In the references, when the propofition referred to is in the fame book with the propofition to be proved, the book is not named, but only the number of the propofition, but, if in any other book, both are named.

THE

ELEMENTS

OF

EUC LI D.

A

BOOK I.

DEFINITION S.

I.

Point is that which hath no parts or magnitude.

II.

A line is length without breadth.

III.

The bounds of a line are points.

IV.

A right line is that which lieth evenly between its points.

V.

A fuperficies is that which hath only length and breadth.

VI.

The bounds of a fuperficies are lines.

VII.

A plain superficies is that which lieth evenly between its lines.

VIII.

A plain angle is the inclination of two lines to one another in the fame plain, which touch each other, but do not lie in the fame right line.

IX.

If the lines containing the angle be right ones, then the angle is called a right-lined angle.

X.

When one right line ftanding on another right line makes the angles on each fide thereof equal to one another, each of these angles is a right one, and that line which ftands upon the other is called a perpendicular to that whereon it ftands.

[blocks in formation]

Book I.

[blocks in formation]

An obtufe angle is that which is greater than a right one.

XII.

An acute angle is that which is less than a right one.

XIII.

A term, or bound, is the extreme of any thing.

XIV.

A figure is that which is contained under one or more terms.

XV.

A circle is a plain figure bounded by one line, called the circumference, to which all right lines drawn from a certain point within the fame are equal.

XVI.

That point is called the center of the circle.

XVII.

The diameter of a circle is a right line drawn through the center, and terminated on both ends by the circumference, and divides the circle into two equal parts.

XVIII.

A femicircle is a figure contained under any diameter, and the circumference cut off by that diameter.

XIX.

A fegment of a circle is a figure contained under a right line, and circumference cut off by that right line.

XX.

Right-lined figures are fuch as are contained by right lines.
XXI.

Three fided figures are fuch as are contained by three lines.
XXII.

Four sided figures are fuch as are contained by four lines.

XXIII.

Many fided figures are fuch as are contained by more than four

lines.

XXIV.

An equilateral triangle is that which hath three equal fides.

XXV.

An ifofceles triangle, that which hath two fides equal.

XXVI.

A fcalene triangle, that which hath all the three fides unequal.
XXVII,

A right angled triangle is that which hath one right angle in it.
XXVIII.

An obtufe angled one, that which hath one obtufe angle in it.

XXIX.

An acute angled triangle is that which hath all the angles lef than right ones.

XXX.

A fquare is that which hath four equal fides, and its angles all right ones.

XXXI.

An oblong, or rectangle, is longer than broad, its oppofite fides are equal, and its angles all right ones.

XXXII.

A rhombus, that which hath four equal fides, but not right angles.

XXXIII.

A rhomboides, whofe oppofite fides and angles are equal.

XXXIV.

All quadrilateral figures beside these are called trapezia.

XXXV.

Parallel right lines are fuch as, being produced both ways in the fame plain, never meet.

XXXVI.

A parallelogram is a figure whose oppofite fides are parallel.

G

POSTULATE S.

I.

RANT that a right line may be drawn from any one
point to another:

II.

That a finite right line may be continued directly forwards: And,

III.

That a circle may be defcribed about any center, with any distance.

Book I.

AXIOM S.

I.

T

HINGS equal to one and the fame thing are equal to
one another.

II.

If equal things are added to equal things, the wholes will be equal.

III.

If from equal things equal things be taken, the remainders will be equal.

IV.

If to unequal things equal things are added, the whole will be

unequal.

V. I

Book I.

V.

If from unequal things equal parts are taken, the remainders will be unequal.

VI.

Things which are double one and the fame thing are equal between themselves.

VII.

Things which are half one and the fame thing are equal between themselves.

VIII.

Things which mutually agree together are equal to one another.

[blocks in formation]

If a right line fall upon two right lines, making the inward angles on the fame fide lefs than two right angles, these right lines continually produced will at last meet one another on that fide where the angles are less than right ones.

N. B. Any angle is expreffed by three letters, of which that at the vertex is named betwixt the other two.

PRO

« ForrigeFortsæt »